CPM Homework Help : APCALC Problem 2-110

Lena loves limits because they help her visualize the graphs of complicated looking functions. For example, by evaluating the limit as as $x \rightarrow \infty$ , she can determine if a rational function such as $f(x) = \frac { p ( x ) } { r ( x ) }$ will have a horizontal asymptote or not.

Evaluate each limit below and explain to Lena how she can determine $\lim\limits _ { x \rightarrow \infty } \frac { p ( x ) } { r ( x ) }$ without graphing. (Be careful! The expressions below are all slightly different.)

$\lim\limits _ { x \rightarrow \infty } ( \frac { x ^ { 2 } - 7 x + 6 } { x ^ { 3 } + 9 x - 2 } )$
Hint (a):
Compare the highest power in the numerator and the denominator small exponent $(x^2)$ on top large exponent $(x^3)$ on the bottom.
More Help (a):
$\lim_{x\rightarrow \infty }\frac{x^{2}}{x^{3}}=\lim_{x\rightarrow \infty }\frac{1}{x}$
Answer (a):
$0$

$\lim\limits _ { x \rightarrow \infty } ( \frac { x ^ { 2 } - 7 x + 6 } { x ^ { 2 } + 9 x - 2 } )$
Hint (b):
As $x \rightarrow \infty$ , only the highest power of the numerator and denominator is powerful enough to affect the end behavior of a function.
More Help (b):
$\lim_{x\rightarrow \infty }\frac{x^{2}}{x^{2}}=$
Answer (b):
$1$

$\lim\limits _ { x \rightarrow \infty } ( \frac { x ^ { 3 } - 7 x + 6 } { x ^ { 2 } + 9 x - 2 } )$
Hint (c):
$\lim_{x\rightarrow \infty }\frac{x^{3}}{x^{2}}=\lim_{x\rightarrow \infty }x=$
Answer (c):
Limit Does Not Exist but $y \rightarrow \infty$